1. Oded Goldreich – Defining Moments
Omer Reingold, Stanford, on OdedFest 2017

2. This Talk’s Concept Focus on Oded’s concepts How does he do it?
And what awesome concepts they are!
Conceptual contributions include everything: the notions, the definitions, the notations …
Even if it doesn’t start with Oded, it often ends with him
How does he do it?
Writing, writing and then some more writing (papers, surveys, books)
His famous personal touch (more, I guess, in the evening sessions)

3. Clarifications and Warnings
Chosen definitions not meant to be representative or Oded’s best (but I love them all)
Bias towards those that have impacted my own research
These are all joint works; There were other papers before them and papers that followed
Celebrating a community
But don’t expect credits (talk to me when its your fest)
Papers contain more than I discuss
And to Oded:
This is your research and your fest
But its my talk! Just saying …

4. 1st Notion: Pseudorandom Functions
Goldreich, Goldwasser and Micali, how to construct random functions, FOCS 84 and JACM 86
The title – such commitment to the computational lens.
I have set up on a Manchester computer a small programme using only 1000 units of storage, whereby the machine supplied with one sixteen figure number replies with another within two seconds. I would defy anyone to learn from these replies sufficient about the programme to be able to predict any replies to untried values.
A. TURING

5. Poly-Random Collections

6. Indistinguishability
g is uniform or in F ? x1
g(x1) g … xt
g(xt)

7. The Uphill Battle
Kolmogorov Complexity: non- constructive and not applicable
Comparison with One-Way functions
Comparison with CSB
(cryptographically strong pseudorandom bit generators)
PRFs vs. simulating random oracle
In particular, allows for sharing a function (distributed)

8. My Connection Learned the definition from Oded’s notes
Editor of two of the journal versions
Some fond memories there
33 years to PRFs + GGM construction and countless papers – no more explanations needed

9. 2nd Notion: Block Sources
Chor and Goldreich, Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity, FOCS 85, SICOMP 88

10. It Contains Everything
Min entropy as THE measure of randomness in a weak random source – X has min-entropy  k if x, Pr[X=x]<2-k
Flat distributions (uniform on 2k elements)
Inner product (Hadamard code) is a two-source extractor for high entropies
Randomized communication complexity, slightly dependent sources, …

11. Block Source

12. My Connection
Constructions of randomness extractors heavily relied on block sources
First extract blocks then extract from the block-source
Zig-zag product analysis measure the entropy in a pair (v,a) of (vertex,edge label), as a block source.

13. 3rd Notion: Property Testing
Goldreich, Goldwasser and Ron, Property Testing and Its Connection to Learning and Approximation, FOCS 96, JACM 98

14. So What’s New? Combinatorial properties
General Distributions, a la PAC learning (Valinat)

15. Since Then
Flourishing and mathematically deep field – the power of a conceptually strong work (and many more that followed)
My connection – PCP composition through a stronger notion, inspired by property testing that we (Dinur and I) called “assignment testers”
PCP proofs allow one to prove that a SAT formula  is satisfiable. An assignment tester allows proving that an assignment  is close to a satisfying assignment of .
Oded et al rejected the gesture and in an independent work called these objects PCPPs
Which stands for “Peace Corps Partnership Program” and “PCPartPicker” and “C99 preprocessor written in pure Python” but also for
Probabilistically Checable Proofs of Proximity
If you can’t beat them join them …

16. 4th Notion: Auxiliary-Input ZK
Goldreich and Oren, Definitions and Properties of Zero- Knowledge Proof Systems. J. Cryptology 1994
Title screams “conceptual”
Zero-Knowledge due to Goldwasser, Micali and Rackoff is a jewel in Cryptography’s crown.
Much of the way we think of ZK was shaped by Oded’s writings - black-box ZK, auxiliary-input ZK, uniform ZK

17. What the Verifier Knows?
I’m convinced x x …
ZK: the verifier doesn’t learn anything (beyond validity)
Auxiliary-input ZK: the verifier doesn’t learn anything new
Vital for composition (either parallel or sequential)

18. Formally …
Both the verifier V* and the simulator MV* have access to the auxiliary-information y

19. My Connection This year’s Gödel Prize winner – Differential Privacy
Definition of privacy in data analysis
What do you learn about a particular row in a database from Differentially-Private analysis?
The definition puts auxiliary-input front and center – even if you know all other rows of the database, you do not learn much about this special row (can’t achieve ZK).
Here too – composition is key
We recently use resilience of DP to composition for better adaptive data analysis

20. Concluding Remarks
Discussed: PRFs, Block Sources, Property Testing, Auxiliary- Input ZK
Wow!
Conceptual contributions are long lasting
What’s next?

21. Happy Birthday